A note on the geometry of figurate numbers

We give a short proof of the formula $n^p=\sum_{\ell=0}^{p-1} (-1)^{\ell} c_{p,\ell} F^{p-\ell}_n$, where $F^{p-\ell}_n$ is the figurate number and $c_{p,\ell}$ is the number of $(p-\ell)$-dimensional facets of $p$-dimensional simplices obtained by cutting the $p$-dimensional cube.

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